Problem: Given $ m \angle RPS = 6x + 17$, and $ m \angle QPR = 2x + 67$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Explanation: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since $\angle QPS$ is a straight angle, we know ${m\angle QPS = 180}$ Substitute in the expressions that were given for each measure: $ {2x + 67} + {6x + 17} = {180}$ Combine like terms: $ 8x + 84 = 180$ Subtract $84$ from both sides: $ 8x = 96$ Divide both sides by $8$ to find $x$ $ x = 12$ Substitute $12$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 6({12}) + 17$ Simplify: $ {m\angle RPS = 72 + 17}$ So ${m\angle RPS = 89}$.